Review Notes for IB Standard Level ... - NLCS Maths Department

Review Notes for IB Standard Level Math

? 2015, Steve Muench steve.muench@

@stevemuench Please feel free to share the link to these notes or my worked solutions to the November 2014 exam

or my worked solutions to the May 2015 (Timezone 2) exam

with any student you believe might benefit from them.

November 3, 2015

1

Contents

1 Algebra

8

1.1 Rules of Basic Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.2 Rules of Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.3 Rules of Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.4 Sequences and Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.5 Arithmetic Sequences and Series . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.6 Sum of Finite Arithmetic Series (u1 + ? ? ? + un) . . . . . . . . . . . . . . . . . . . 9 1.7 Partial Sum of Finite Arithmetic Series (uj + ? ? ? + un) . . . . . . . . . . . . . . . 10

1.8 Geometric Sequences and Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.9 Sum of Finite Geometric Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.10 Sum of Infinite Geometric Series . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.10.1 Example Involving Sum of Infinite Geometric Series . . . . . . . . . . . . 11

1.11 Sigma Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.11.1 Sigma Notation for Arithmetic Series . . . . . . . . . . . . . . . . . . . . . 12

1.11.2 Sigma Notation for Geometric Series . . . . . . . . . . . . . . . . . . . . . 13

1.11.3 Sigma Notation for Infinite Geometric Series . . . . . . . . . . . . . . . . 13

1.11.4 Defining Functions Using Sigma Notation . . . . . . . . . . . . . . . . . . 14

1.12 Applications: Compound Interest . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.13 Applications: Population Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.14 Logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.15 Using Logarithms to Solve Equations . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.16 Using Exponentiation to Solve Equations . . . . . . . . . . . . . . . . . . . . . . 16

1.17 Logarithm Facts Involving 0 and 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.18 Laws of Exponents and Logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.19 Change of Base . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.20 Powers of Binomials and Pascal's Triangle . . . . . . . . . . . . . . . . . . . . . . 17 1.21 Expansion of (a + b)n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.22 The Binomial Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.22.1 Using The Binomial Theorem for a Single Term . . . . . . . . . . . . . . . 21

1.22.2 Example of Using Binomial Theorem . . . . . . . . . . . . . . . . . . . . . 21

1.23 Solving Systems of Three Linear Equations Using Substitution . . . . . . . . . . 23

1.24 Solving Systems of Three Linear Equations Using Matrix Inverse . . . . . . . . . 23

1.24.1 Writing a System of Linear Equations in Matrix Form . . . . . . . . . . . 24

1.24.2 Using Matrix Inverse to Solve System of Three Linear Equations . . . . . 24

2 Functions and Equations

26

2.1 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.2 Union and Intersection of Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.3 Common Sets of Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.4 Intervals of Real Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.5 Concept of Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.6 Graph of a Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.7 Domain of a Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.8 Range of a Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.9 Composing one Function with Another . . . . . . . . . . . . . . . . . . . . . . . . 30

2.10 Identity Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.11 Inverse Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.12 Determining the Inverse Function as Reflection in Line y = x . . . . . . . . . . . 31

2.13 Determining the Inverse Function Analytically . . . . . . . . . . . . . . . . . . . 32

2

2.14 Drawing and Analyzing Graphs with Your Calculator . . . . . . . . . . . . . . . 33 2.14.1 Drawing and Zooming Graph . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.14.2 Finding a Maximum Value in an Interval . . . . . . . . . . . . . . . . . . 34 2.14.3 Finding a Minimum Value Value in an Interval . . . . . . . . . . . . . . . 34 2.14.4 Finding the x-Intercepts or "Zeros" of a Graph in an Interval . . . . . . . 35 2.14.5 Finding the y-Intercept of a Graph . . . . . . . . . . . . . . . . . . . . . . 35 2.14.6 Vertical Asymptotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.14.7 Graphing Vertical Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.14.8 Horizontal Asymptotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.14.9 Tips to Compute Horizontal Asymptotes of Rational Functions . . . . . . 37 2.14.10 Graphing Horizontal Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.14.11 Symmetry: Odd Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.14.12 Symmetry: Even Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.14.13 Solving Equations Graphically . . . . . . . . . . . . . . . . . . . . . . . . 39

2.15 Transformations of Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.15.1 Horizontal and Vertical Translations . . . . . . . . . . . . . . . . . . . . . 40 2.15.2 Reflections in x-Axis and y-Axis . . . . . . . . . . . . . . . . . . . . . . . 41 2.15.3 Vertical Stretch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 2.15.4 Horizontal Stretch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 2.15.5 Order Matters When Doing Multiple Transformations in Sequence . . . . 43 2.15.6 Graphing the Result of a Sequence of Transformations . . . . . . . . . . . 44 2.15.7 Determining Where a Particular Point Moves Under a Sequence of Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

2.16 Quadratic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 2.16.1 Using the Quadratic Formula to Find Zeros of Quadratic Function . . . . 46 2.16.2 Finding the Vertex If You Know the Zeros . . . . . . . . . . . . . . . . . . 47 2.16.3 Graph and Axis of Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . 47 2.16.4 Computing the Vertex From the Coefficients . . . . . . . . . . . . . . . . 48 2.16.5 Using the Discriminant to Find the Number of Zeros . . . . . . . . . . . . 49 2.16.6 Y Intercept Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 2.16.7 X Intercept Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 2.16.8 Binomial Squared Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 2.16.9 Vertex (h, k) Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

2.17 Reciprocal Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 2.18 Rational Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 2.19 Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 2.20 Continuously Compounded Interest . . . . . . . . . . . . . . . . . . . . . . . . . . 55 2.21 Continuous Growth and Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 2.22 Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3 Circular Functions and Trigonometry

57

3.1 Understanding Radians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.1.1 Degrees Represent a Part of a Circular Path . . . . . . . . . . . . . . . . . 57

3.1.2 Computing the Fraction of a Complete Revolution an Angle Represents . 57

3.1.3 Attempting to Measure an Angle Using Distance . . . . . . . . . . . . . . 57

3.1.4 Arc Distance on the Unit Circle Uniquely Identifies an Angle . . . . . . 59

3.1.5 Computing the Fraction of a Complete Revolution for Angle in Radians . 59

3.2 Converting Between Radians and Degrees . . . . . . . . . . . . . . . . . . . . . . 59

3.2.1 Converting from Degrees to Radians . . . . . . . . . . . . . . . . . . . . . 59

3.2.2 Converting from Radians to Degrees . . . . . . . . . . . . . . . . . . . . . 60

3.3 Length of an Arc Subtended by an Angle . . . . . . . . . . . . . . . . . . . . . . 60

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3.4 Inscribed and Central Angles that Subtend the Same Arc . . . . . . . . . . . . . 61 3.5 Area of a Sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.6 Definition of cos and sin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 3.7 Interpreting cos and sin on the Unit Circle . . . . . . . . . . . . . . . . . . . . 63 3.8 Radian Angle Measures Can Be Both Positive and Negative . . . . . . . . . . . . 63 3.9 Remembering the Exact Values of Key Angles on Unit Circle . . . . . . . . . . . 64 3.10 The Pythagorean Identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.11 Double Angle Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 3.12 Definition of tan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 3.13 Using a Right Triangle to Solve Trigonometric Problems . . . . . . . . . . . . . . 66

3.13.1 Using Right Triangle with an Acute Angle . . . . . . . . . . . . . . . . . . 66 3.13.2 Using Right Triangle with an Obtuse Angle . . . . . . . . . . . . . . . . . 67 3.14 Using Inverse Trigonometric Functions on Your Calculator . . . . . . . . . . . . . 68 3.15 Circular Functions sin, cos, and tan . . . . . . . . . . . . . . . . . . . . . . . . . 69 3.15.1 The Graph of sin x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 3.15.2 The Graph of cos x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 3.15.3 The Graph of tan x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 3.15.4 Transformations of Circular Functions . . . . . . . . . . . . . . . . . . . . 70 3.15.5 Using Transformation to Highlight Additional Identities . . . . . . . . . . 71 3.15.6 Determining Period from Minimum and Maximum . . . . . . . . . . . . . 72 3.16 Applications of the sin Function: Tide Example . . . . . . . . . . . . . . . . . . . 72 3.17 Solving Trigonometric Equations in a Finite Interval . . . . . . . . . . . . . . . . 74 3.18 Solving Quadratic Equations in sin, cos, and tan . . . . . . . . . . . . . . . . . . 74 3.19 Solutions of Right Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 3.20 The Cosine Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 3.21 The Sine Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 3.22 Area of a Triangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4 Vectors

78

4.1 Vectors as Displacements in the Plane . . . . . . . . . . . . . . . . . . . . . . . . 78

4.2 Vectors as Displacements in Three Dimensions . . . . . . . . . . . . . . . . . . . 78

4.3 Terminology: Tip and Tail . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

4.4 Representation of Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.5 Magnitude of a Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.6 Multiplication of a Vector by a Scalar . . . . . . . . . . . . . . . . . . . . . . . . 81

4.7 Negating a Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

4.8 Sum of Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4.9 Difference of Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4.10 Unit Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.11 Scaling Any Vector to Produce a Parallel Unit Vector . . . . . . . . . . . . . . . 84

4.12 Position Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

4.13 Determining Whether Vectors are Parallel . . . . . . . . . . . . . . . . . . . . . . 85

4.14 Finding Parallel Vector with Certain Fixed Length . . . . . . . . . . . . . . . . . 86

4.15 Scalar (or "Dot") Product of Two Vectors . . . . . . . . . . . . . . . . . . . . . . 86

4.16 Perpendicular Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

4.17 Base Vectors for Two Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

4.18 Base Vectors for Three Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . 87

4.19 The Angle Between Two Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

4.20 Vector Equation of a Line in Two and Three Dimensions . . . . . . . . . . . . . . 88

4.21 Vector Equation of Line Passing Through Two Points . . . . . . . . . . . . . . . 89

4.22 Finding the Cartesian Equation from a Vector Line . . . . . . . . . . . . . . . . . 90

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4.23 The Angle Between Two Vector Lines . . . . . . . . . . . . . . . . . . . . . . . . 91 4.24 Distinguishing Between Coincident and Parallel Lines . . . . . . . . . . . . . . . 91 4.25 Finding the Point of Intersection of Two Lines . . . . . . . . . . . . . . . . . . . 91

4.25.1 Finding Intersection Between a Line and an Axis . . . . . . . . . . . . . . 92

5 Statistics

94

5.1 Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

5.1.1 Population versus Sample . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

5.1.2 Discrete Data versus Continuous Data . . . . . . . . . . . . . . . . . . . . 94

5.2 Presentation of Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

5.2.1 Frequency Distribution Tables . . . . . . . . . . . . . . . . . . . . . . . . 94

5.2.2 Frequency Histograms with Equal Class Intervals . . . . . . . . . . . . . . 95

5.3 Mean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

5.4 Median . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

5.5 Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

5.6 Cumulative Frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

5.6.1 Cumulative Frequency Table . . . . . . . . . . . . . . . . . . . . . . . . . 97

5.6.2 Cumulative Frequency Graphs . . . . . . . . . . . . . . . . . . . . . . . . 97

5.7 Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

5.7.1 Range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

5.7.2 Quartiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

5.7.3 Interquartile Range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

5.7.4 Box and Whiskers Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

5.7.5 Percentiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

5.7.6 Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

5.7.7 Standard Deviation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

5.8 Use Cumulative Frequency Graph to Find Median and Quartiles . . . . . . . . . 100

5.9 Computing Statistics Using Your Calculator . . . . . . . . . . . . . . . . . . . . . 101

5.9.1 Calculating Statistics for a Single List . . . . . . . . . . . . . . . . . . . . 101

5.9.2 Calculating Statistics for a List with Frequency . . . . . . . . . . . . . . . 102

5.10 Linear Correlation of Bivariate Data . . . . . . . . . . . . . . . . . . . . . . . . . 102

5.10.1 Scatter Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

5.10.2 Pearson's Product-Moment Correlation Coefficient r . . . . . . . . . . . . 103

5.10.3 Lines of Best Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

5.10.4 Equation of the Regression Line . . . . . . . . . . . . . . . . . . . . . . . 104

6 Probability

106

6.1 Probability Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

6.2 Probability of an Event . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

6.3 Probability of an Event's Not Occurring . . . . . . . . . . . . . . . . . . . . . . . 106

6.4 Independent, Dependent, and Mutually Exclusive Events . . . . . . . . . . . . . 106

6.5 Probability of A and B for Independent Events . . . . . . . . . . . . . . . . . . . 107

6.6 Probability of A or B for Mutually Exclusive Events . . . . . . . . . . . . . . . . 108

6.7 Probability of Mutually Exclusive Events . . . . . . . . . . . . . . . . . . . . . . 108

6.8 Probability of A or B for Non-Mutually Exclusive Events . . . . . . . . . . . . . 108

6.9 Lists and Tables of Outcomes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

6.10 Conditional Probability of Dependent Events . . . . . . . . . . . . . . . . . . . . 109

6.11 Testing for Independent Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

6.12 Venn Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

6.13 Tree Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

6.14 Probabilities With and Without Replacement . . . . . . . . . . . . . . . . . . . . 110

5

6.15 Discrete Random Variables and Their Probability Distributions . . . . . . . . . . 111 6.15.1 Explicitly Listed Probabilities . . . . . . . . . . . . . . . . . . . . . . . . . 111 6.15.2 Probability Distribution Given by a Function . . . . . . . . . . . . . . . . 111 6.15.3 Explicit Probability Distribution Involving an Unknown . . . . . . . . . . 112

6.16 Expected Value (Mean), E(X) for Discrete Data . . . . . . . . . . . . . . . . . . 113 6.16.1 Expected Value for a "Fair" Game . . . . . . . . . . . . . . . . . . . . . . 114

6.17 Binomial Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 6.17.1 Using the Calculator for Binomial Distribution Problems . . . . . . . . . 115 6.17.2 Tip for Calculators Whose binomCdf Does Not Have Lower Bound . . . . 116

6.18 Mean and Variance of the Binomial Distribution . . . . . . . . . . . . . . . . . . 117 6.18.1 Example of Mean and Variance of Binomial Distribution . . . . . . . . . . 117

6.19 Normal Distribution and Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 6.20 Standardizing Normal Variables to Get z-values (z-scores) . . . . . . . . . . . . . 118 6.21 Using the Calculator for Normal Distribution Problems . . . . . . . . . . . . . . 119

6.21.1 Tip for Calculators Whose normalCdf Does Not Have Lower Bound . . . 119 6.22 Using Inverse Normal Cumulative Density Function . . . . . . . . . . . . . . . . 120 6.23 Determining z Value and from the Probability . . . . . . . . . . . . . . . . . . 120

7 Calculus

122

7.1 Overview of Concepts for Differential Calculus . . . . . . . . . . . . . . . . . . . 122

7.1.1 Rate of Change in Distance . . . . . . . . . . . . . . . . . . . . . . . . . . 122

7.1.2 Function's Derivative Gives Instantaneous Rate of Change Using Tangent

Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

7.1.3 Relationships Between Function and Derivative . . . . . . . . . . . . . . . 125

7.1.4 Summary of Derivative Concepts . . . . . . . . . . . . . . . . . . . . . . . 127

7.2 Equations of Tangents and Normals . . . . . . . . . . . . . . . . . . . . . . . . . 127

7.3 Notation for Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

7.4 Graphing Derivatives with Your Calculator . . . . . . . . . . . . . . . . . . . . . 128

7.4.1 Graphing Derivatives on the TI-Nspire CX . . . . . . . . . . . . . . . . . 129

7.4.2 Graphing Derivatives on the TI-84 Silver Edition . . . . . . . . . . . . . . 130

7.5 Computing Derivatives at a Point with Your Calculator . . . . . . . . . . . . . . 130

7.5.1 Computing Derivative at a Point on the TI-NSpire CX . . . . . . . . . . . 130

7.5.2 Computing Derivative at a Point on the TI-84 Silver Edition . . . . . . . 131

7.6 Rules for Computing Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

7.6.1 Derivative of xn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

7.6.2 Derivative of sin x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

7.6.3 Derivative of cos x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

7.6.4 Derivative of tan x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

7.6.5 Derivative of ex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

7.6.6 Derivative of ln x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

7.7 Differentiating a Scalar Multiple . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

7.8 Differentiating a Sum or Difference . . . . . . . . . . . . . . . . . . . . . . . . . . 133

7.9 Example Using Derivation Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

7.10 "Chain Rule": Differentiating Composed Functions . . . . . . . . . . . . . . . . . 133

7.11 "Product Rule": Differentiating Product of Functions . . . . . . . . . . . . . . . 135

7.12 "Quotient Rule": Differentiating Quotient of Functions . . . . . . . . . . . . . . . 138

7.13 Using the First Derivative to Find Local Maxima and Minima . . . . . . . . . . . 140

7.14 Analyzing Zeros of Derivative Graph to Find Maxima and Minima . . . . . . . . 141

7.15 Using the Second Derivative to Determine Concavity and Points of Inflection . . 142

7.16 Overview of Concepts for Integral Calculus . . . . . . . . . . . . . . . . . . . . . 142

7.16.1 Area Under Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

6

7.16.2 Relationship Between Derivation and Integration . . . . . . . . . . . . . . 144

7.17 Anti-Differentiation to Compute Integrals . . . . . . . . . . . . . . . . . . . . . . 145

7.17.1 Indefinite Integration as Anti-Differentiation . . . . . . . . . . . . . . . . 146

7.18 Rules for Computing the Indefinite Integral (Antiderivative) . . . . . . . . . . . . 146

7.18.1 Indefinite Integral of xn (n Q) . . . . . . . . . . . . . . . . . . . . . . . 147

7.18.2 Indefinite Integral of sin x . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

7.18.3 Indefinite Integral of cos x . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

7.18.4

Indefinite

Integral

of

1 x

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

148

7.18.5 Indefinite Integral of ex . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

7.19 Integrating Constant Multiple of a Function . . . . . . . . . . . . . . . . . . . . . 148

7.20 Integrating Sums and Differences of Functions . . . . . . . . . . . . . . . . . . . . 149

7.21 Integration Using Substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

7.21.1 Simple Example Involving Linear Factor . . . . . . . . . . . . . . . . . . . 150

7.21.2 Example Involving Additional Factor of x . . . . . . . . . . . . . . . . . . 151

7.21.3 Example Involving Other Additional Factors . . . . . . . . . . . . . . . . 152

7.21.4 Summary of Strategy for Integration by Substitution . . . . . . . . . . . . 152

7.22 Using Additional Information to Determine Constant of Integration . . . . . . . 152

7.23 Computing the Definite Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

7.23.1 Computing the Definite Integral Analytically . . . . . . . . . . . . . . . . 154

7.23.2 Computing the Definite Integral Using Technology . . . . . . . . . . . . . 155

7.24 Displacement s, Velocity v, and Acceleration a . . . . . . . . . . . . . . . . . . . 155

7.25 Determining Position Function from Acceleration . . . . . . . . . . . . . . . . . . 156

7.26 Total Area Under a Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

7.27 Area Between Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

7.27.1 Area Between Curves Using Calculator . . . . . . . . . . . . . . . . . . . 157

7.27.2 Area Between Curves Analytically (Without the Calculator) . . . . . . . 158

7.28 Net Change in Displacement versus Total Distance Traveled . . . . . . . . . . . . 158

7.28.1 Difference Between Distance and Displacement . . . . . . . . . . . . . . . 158

7.28.2 Computing Total Distance Traveled . . . . . . . . . . . . . . . . . . . . . 159

7

1 Algebra

1.1 Rules of Basic Operations

You need to be familiar with the rules of basic operations shown in Table 1.

a (b + c) = ab + ac

a b

+

c d

=

a b

?

d d

+

c d

?

b b

=

ad+cb bd

- (a - b) = b - a

(a - b)2 = a2 - 2ab + b2

a b

?

c d

=

ac bd

- (a + b) = -a - b

(a + b)2 = a2 + 2ab + b2

(a + b) (a - b) = a2 - b2

(a + b) (c + d)

Table 1: Basic Operations

1.2 Rules of Roots

You need to be familiar with the rules of roots shown in Table 2.

ab = a b

a-

p q

=

( q1a)p

a

b

p a

?

a a

=

pa a

1 aq = q a (a)2 = a

p

aq

=

(q a)p

(q a)q = a

Table 2: Rules of Roots

1.3 Rules of Exponents

You need to be familiar with the rules of exponents shown in Table 3.

ab ? ac = ab+c

ab c = ab?c

a d

c

=

ac dc

ab ac

= a(b-c)

a(-c)

=

1 ac

ab de

?

dc af

=

ab af

?

dc de

= ab-f

? dc-e

=

ab-f de-c

Table 3: Rules of Exponents

1.4 Sequences and Series

A sequence is an ordered list of numbers. The list can be finite or infinite. We name the sequence with a letter like u, and refer to an individual element (or "term") of this ordered list using a subscript that represents the ordinal position the particular element occupies in the list. The first term in the sequence u is denoted by u1, the second term by u2, and so on. The nth term in the sequence is un.

You describe a sequence by giving a formula for its nth term. This could be something simple like the sequence each of whose terms is the number 1:

un = 1

8

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